
For low \(\gamma \)  (ELS limit), both \(P(s)\)  and \(Q(E)\)  are scale-free distributions with an universal exponent \(-2.5\) . This is shown in figure REF (a) and (b) with \(\gamma =0.1\) . This is due to the fact that in this limit, \(\langle E \rangle \sim s\) , and the distribution of \(E\)  should be equal to the distribution \(s\)  which is a scale-free distribution with exponent \(-2.5\)  in the ELS limit [1]}.

On the other hand, when \(\gamma \)  is very high \(Q(E)\)  is a scale-free distribution in spite of the fact that the distribution \(P(s)\)  is an exponential [2]} here (shown in figure REF (a) and (b) with \(\gamma =8.0\) ). This is due to the lack of correlation between \(s\)  and \(E\)  which is presented in figure REF  and REF .

For an intermediate \(\gamma \)  value, both \(P(s)\)  and \(Q(E)\)  are exponential distributions. The possible reason could be the fact that the \(\gamma \)  value where \(P(s)\)  changes from power law to exponential is not the same \(\gamma \)  value where the correlation between \(s\)  and \(E\)  decreases drastically. This will be clear when we will quantitatively discuss the correlation function. Then there will be an window of \(\gamma \)  where the correlation is high but \(P(s)\)  is exponential, producing an exponential distribution of \(Q(E)\) .

